ISA 444/544: Business Forecasting

class: center, middle, inverse, title-slide

.title[
# ISA 444/544: Business Forecasting
]
.subtitle[
## 06: Centered Moving Average and Seasonal Decomposition
]
.author[
### <br>Fadel M. Megahed, PhD <br><br> Raymond E. Glos Professor in Business <br> Farmer School of Business<br> Miami University<br><br> <a href="https://twitter.com/FadelMegahed"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" /></svg> <span class="citation">@FadelMegahed</span></a> <br> <a href="https://github.com/fmegahed/"><svg viewBox="0 0 496 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" /></svg> fmegahed</a> <br> <a href="mailto:fmegahed@miamioh.edu"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z" /></svg> fmegahed@miamioh.edu</a><br> <a href="https://calendly.com/fmegahed"><svg viewBox="0 0 384 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M202.021 0C122.202 0 70.503 32.703 29.914 91.026c-7.363 10.58-5.093 25.086 5.178 32.874l43.138 32.709c10.373 7.865 25.132 6.026 33.253-4.148 25.049-31.381 43.63-49.449 82.757-49.449 30.764 0 68.816 19.799 68.816 49.631 0 22.552-18.617 34.134-48.993 51.164-35.423 19.86-82.299 44.576-82.299 106.405V320c0 13.255 10.745 24 24 24h72.471c13.255 0 24-10.745 24-24v-5.773c0-42.86 125.268-44.645 125.268-160.627C377.504 66.256 286.902 0 202.021 0zM192 373.459c-38.196 0-69.271 31.075-69.271 69.271 0 38.195 31.075 69.27 69.271 69.27s69.271-31.075 69.271-69.271-31.075-69.27-69.271-69.27z" /></svg> Automated Scheduler for Office Hours</a><br><br>
]
.date[
### Fall 2025
]

---

## Quick Refresher of Last Class

✅ Generate and interpret simple line charts

✅ Create seasonal plots and subplots

✅ Describe a lag and create lag scatter plots

✅ Plot and interpret the autocorrelation function (ACF) for a time-series

---

## Learning Objectives for Today's Class

- Describe and compute centered moving averages

- Estimate trend-cycle via moving averages

- Perform classical decomposition (trend-cycle, seasonal, residual/remainder)

- Understand STL / MSTL as alternatives to classical decomposition

---
class: inverse, center, middle

# Motivation and Context

---

## Recall: Time Series Components

- **Trend**: Long-term increase or decrease in the data

- **Seasonal**: Regular pattern of up and down fluctuations influenced by calendar or time of day (e.g., quarterly, monthly, day of the week, hourly)

- **Cyclical**: Repeating up and down movements that are **not of a fixed period** (e.g., business cycles, economic cycles), typically over multiple years

---

## Time Series Decomposition

- **Decomposition** is the process of breaking down a time series into its constituent components: trend-cycle, seasonal, and residual/remainder.

- **Mathematically**, we can write a time series `$y_t$` as a function of these components: `$$y_t = f(T_t, \ S_t, \ R_t),$$` where:  
  + `$T_t$`: Trend-cycle component  
  + `$S_t$`: Seasonal component    
  + `$R_t$`: Remainder component
  + `$f()$`: Some function that combines the components

- The decomposition can be **additive** or **multiplicative**.
  
---

## Seasonality: Additive Models

**Definition:** An additive model is appropriate when the **trend is approximately linear**, and the **seasonal components stays constant over time.**   
It can be written as: `$y_t = T_t + S_t + R_t$`.

---

## Seasonality: Multiplicative Models

**Definition:** A multiplicative model is appropriate when the **trend is nonlinear (e.g., exponential)**, and/or the **seasonal component changes proportionally with the level of the time series.**   
It can be written as: `$y_t = T_t \times S_t \times R_t$`.

---

## Motivating Example: US Retail Trade Employment

.panelset[

.panel[.panel-name[Data]
- The [data](https://fred.stlouisfed.org/series/CEU4200000001) contains the **total employment in U.S. retail trade** sector from 1992 to 2025.

- The [data](https://fred.stlouisfed.org/series/CEU4200000001) is **monthly** and **not seasonally adjusted**.

]

.panel[.panel-name[Plot]

]

.panel[.panel-name[Decomp]

.font80[

.pull-left[
- **Decomposition** is the process of breaking down a time series into its constituent components: trend-cycle, seasonal, and residual/remainder.

- Here, I am using the **classical decomposition method to decompose** the retail trade employment time series into its components.

- We will talk about how that is done **later**.

- What I want you to focus on is **understanding the charts** and the following **concepts**:  
  + Trend-Cycle  
  + DeTrended Series  
  + Seasonally Adjusted Series
]

.pull-right[

|Date       |   `$y_t$`|   `$T_t$`|  `$S_t$`|  `$R_t$`|
|:----------|-------:|-------:|------:|------:|
|1992-01-01 | 12784.3|      NA|  -61.7|     NA|
|1992-02-01 | 12581.9|      NA| -238.6|     NA|
|1992-03-01 | 12543.3|      NA| -220.0|     NA|
|1992-04-01 | 12632.0|      NA| -232.7|     NA|
|1992-05-01 | 12735.4|      NA| -125.1|     NA|
|1992-06-01 | 12819.7|      NA|  -17.3|     NA|
|1992-07-01 | 12799.8| 12828.3|  -12.0|  -16.5|
|1992-08-01 | 12805.1| 12835.5|   -9.5|  -20.9|
|1992-09-01 | 12778.3| 12844.6|  -92.0|   25.7|
|1992-10-01 | 12884.5| 12853.4|   41.7|  -10.6|
|1992-11-01 | 13141.0| 12862.8|  401.8| -123.6|
]
]
]

.panel[.panel-name[Decomp]

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/decompose_retail_emp2-1.png" width="100%" style="display: block; margin: auto;" />
]

.panel[.panel-name[Trend-Cyc]

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/tc_overlay_retail_emp-3.png" width="100%" style="display: block; margin: auto;" />
]

.panel[.panel-name[DeTrend]

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/detrend_retail_emp-5.png" width="100%" style="display: block; margin: auto;" />
.center[.font50[**Note:** The detrended series is obtained by subtracting the trend-cycle component from the observed TS, i.e., `$y_t - T_t$`.]]
]

.panel[.panel-name[SeasAdj]

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/deseasonalize_retail_emp-7.png" width="100%" style="display: block; margin: auto;" />
.center[.font50[**Note:** The deasonalized series is obtained by subtracting the seasonal component from the observed TS `$y_t - S_t$`. It is similar to the seasonally adjusted [USTRADE](https://fred.stlouisfed.org/series/USTRADE) series.]]
]

.panel[.panel-name[Notes]

.can-edit.key-activity5[

- Edit me  
  
- ... 
  
- ...

- ...

]
]
]

???
From <https://otexts.com/fpp3/components.html>
Seasonally adjusted series contain the remainder component as well as the trend-cycle. Therefore, they are not “smooth”, and “downturns” or “upturns” can be misleading. If the purpose is to look for turning points in a series, and interpret any changes in direction, then it is better to use the trend-cycle component rather than the seasonally adjusted data.

---
class: inverse, center, middle

# Centered Moving Averages

---

## Relation of Moving Averages to Classical Decomposition

- The **classical method of time series decomposition** originated in the 1920s and was widely used until the 1950s.

- It still **forms the basis of many time series decomposition methods**, so it is important to understand how it works.

- The **first step in a classical decomposition is to use a moving average method to estimate the trend-cycle**, so we begin by discussing moving averages.

.footnote[
<html>
<hr>
</html>

**Source:** Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. <https://otexts.com/fpp3/>. Accessed on February 12, 2025. [Chapter 3.3](https://otexts.com/fpp3/moving-averages.html) 
]

---

## Moving Average (MA) Smoothing

- **Moving averages** are used to **smooth out short-term fluctuations** and highlight longer-term trends or cycles.

- In classical decomposition, the **trend-cycle component is estimated using a centered moving average**.

- A **centered moving average** is calculated by taking the average of values on either side of the data point in question.

- Mathematically,a centered moving average of order `$m$` can be written as: `$$\hat{T}_{t} = \frac{1}{m} \sum_{j=-k}^k y_{t+j},$$`
where `$m = 2k + 1$` is the number of values in the moving average.

---

## Activity: Centered Moving Average

Let's consider the following series:  `$y_t = \{4, 7, 7, 10, 13, 13, 16\}$`

Calculate a **centered moving average of order 3**, and **input it in the CMA3 column below.**    
  + Note each **cell** in the table is **editable**, but please **only edit the CMA3 column**.

.font80[
<div class="datatables html-widget html-fill-item" id="htmlwidget-22c625fb4c9951a71ad9" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-22c625fb4c9951a71ad9">{"x":{"filter":"none","vertical":false,"editable":{"target":"cell","disable":{"columns":null},"numeric":[0,1],"date":[]},"data":[[1,2,3,4,5,6,7],[4,7,7,10,13,13,16],[null,null,null,null,null,null,null]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>t<\/th>\n      <th>y_t<\/th>\n      <th>cma3<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":7,"dom":"t","autoWidth":false,"columnDefs":[{"className":"dt-right","targets":[0,1]},{"name":"t","targets":0},{"name":"y_t","targets":1},{"name":"cma3","targets":2}],"order":[],"orderClasses":false,"lengthMenu":[7,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":[],"jsHooks":[]}</script>
]

---

## Centered Moving Averages in Python

``` python
y_t = [4, 7, 7, 10, 13, 13, 16]

df = (
  pd.DataFrame({"t": range(1, len(y_t) + 1), "y_t": y_t})
  .assign(
    cma3 = lambda x: x['y_t'].rolling(window = 3, center = True).mean(),
    cma4r = lambda x: x['y_t'].rolling(window = 4, center = True).mean(),
    cma4l = lambda x: x['cma4r'].shift(-1)
    ) 
  )
df.head(7)
```

```
##    t  y_t  cma3  cma4r  cma4l
## 0  1    4   NaN    NaN    NaN
## 1  2    7   6.0    NaN   7.00
## 2  3    7   8.0   7.00   9.25
## 3  4   10  10.0   9.25  10.75
## 4  5   13  12.0  10.75  13.00
## 5  6   13  14.0  13.00    NaN
## 6  7   16   NaN    NaN    NaN
```

**Which of the two CMA4 cols is correct?** .can-edit.key-activity6[Edit me]

---

## Centered Moving Averages as Trend-Cycle Estimators

---

## Moving Average of Moving Averages

When `$m$` is even, we often resort to using a **moving average of moving averages** to center the moving average. Let us revisit our previous example:

.font90[

``` python
df = (
  pd.DataFrame({"t": range(1, len(y_t) + 1), "y_t": y_t})
  .assign(
    cma3 = lambda x: x['y_t'].rolling(window = 3, center = True).mean(),
    cma4r = lambda x: x['y_t'].rolling(window = 4, center = True).mean(),
    cma4l = lambda x: x['cma4r'].shift(-1),
*   cma4_2 = lambda x: x['cma4l'].rolling(window = 2, center = True).mean(),
*   cma4_2_alt = lambda x: (x['cma4r'] + x['cma4l'])/2
    ) 
  )
df.head(7)
```

```
##    t  y_t  cma3  cma4r  cma4l  cma4_2  cma4_2_alt
## 0  1    4   NaN    NaN    NaN     NaN         NaN
## 1  2    7   6.0    NaN   7.00     NaN         NaN
## 2  3    7   8.0   7.00   9.25   8.125       8.125
## 3  4   10  10.0   9.25  10.75  10.000      10.000
## 4  5   13  12.0  10.75  13.00  11.875      11.875
## 5  6   13  14.0  13.00    NaN     NaN         NaN
## 6  7   16   NaN    NaN    NaN     NaN         NaN
```
]

---

## Moving Average of Moving Averages (Cont.)

When a 2-MA follows a moving average of an even order (such as 4), it is called a “centered moving average of order 4”.

This is because the results are now symmetric.

To see that this is the case, we can write the `$2\times 4$`-MA (named as `cma4_2` or `cma4_2_alt` in our Python code) as follows:

$$
`\begin{align*}
  \hat{T}_{t} &= \frac{1}{2}\Big[
    \frac{1}{4} (y_{t-2}+y_{t-1}+y_{t}+y_{t+1}) +
    \frac{1}{4} (y_{t-1}+y_{t}+y_{t+1}+y_{t+2})\Big] \\
             &= \frac{1}{8}y_{t-2}+\frac14y_{t-1} +
             \frac14y_{t}+\frac14y_{t+1}+\frac18y_{t+2}.
\end{align*}`
$$

It is now a **weighted average of observations that is symmetric**.

.footnote[
<html>
<hr>
</html>

---

## Centered Moving Averages for Trend-Cycle Estimation

📊 **Key Idea:**  
Centered moving averages (CMAs) help estimate the **trend-cycle** by smoothing seasonal fluctuations.

<br>

**Example: 2 × 4 - MA**

`$$\hat{T}_t = \frac{1}{8}y_{t-2} + \frac{1}{4}y_{t-1} + \frac{1}{4}y_t + \frac{1}{4}y_{t+1} + \frac{1}{8}y_{t+2}$$`
<br>

✅ **Balances** data across  quarters  
✅ **Averages out seasonal /quarterly variation**

---

## Choosing the Right Moving Average

| **Seasonal Period (m)** | **Recommended MA** |
|----------------|----------------|
| **Even** (e.g., 12 months, 4 quarters) | **2 × m - MA** |
| **Odd** (e.g., 7 days) | **m-MA** |

<br>

✔ **2 × 12 - MA** for monthly seasonality in annual data  
✔ **2 × 4 - MA** for quarterly seasonality in annual data  
✔ **7-MA** for weekly seasonality in daily data

<br>

⚠ **Beware!**  
Using the wrong MA order **retains seasonal noise** instead of extracting the trend.

---

## Activity: Trend-Cycle Estimation Using CMAs

.panelset[

.panel[.panel-name[Data]

- Download the [CEU4200000001](https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23ebf3fb&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1320&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=USTRADE&scale=left&cosd=1992-01-01&coed=2025-01-01&line_color=%230073e6&link_values=false&line_style=solid&mark_type=none&mw=3&lw=3&ost=-99999&oet=99999&mma=0&fml=a&fq=Monthly&fam=avg&fgst=lin&fgsnd=2020-02-01&line_index=1&transformation=lin&vintage_date=2025-02-12&revision_date=2025-02-12&nd=1939-01-01), capturing the **total employment in U.S. retail trade** sector from 1992 to 2025.

- The data is **monthly** and **not seasonally adjusted**.

- Use an **appropriate moving average** to estimate the trend-cycle component.

- Recreate the plot of raw series and the trend-cycle component from the next tab.

]

.panel[.panel-name[Plot]

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/retail_empl_cma-3.png" width="100%" style="display: block; margin: auto;" />
]

.panel[.panel-name[Printout]

.font80[

```
##            date CEU4200000001      month     cmar     cmal     cma_2
## 0    1992-01-01       12784.3    January      NaN      NaN       NaN
## 1    1992-02-01       12581.9   February      NaN      NaN       NaN
## 2    1992-03-01       12543.3      March      NaN      NaN       NaN
## 3    1992-04-01       12632.0      April      NaN      NaN       NaN
## 4    1992-05-01       12735.4        May      NaN      NaN       NaN
## 5    1992-06-01       12819.7       June      NaN  12825.9       NaN
## 6    1992-07-01       12799.8       July  12825.9  12830.8  12828.35
## 7    1992-08-01       12805.1     August  12830.8  12840.2   12835.5
## 8    1992-09-01       12778.3  September  12840.2  12848.9  12844.55
## 9    1992-10-01       12884.5    October  12848.9  12857.8  12853.35
## ...         ...           ...        ...      ...      ...       ...
## 387  2024-04-01       15406.9      April  15536.4  15534.6   15535.5
## 388  2024-05-01       15487.1        May  15534.6  15532.0   15533.3
## 389  2024-06-01       15557.3       June  15532.0  15532.0   15532.0
## 390  2024-07-01       15540.0       July  15532.0  15536.1  15534.05
## 391  2024-08-01       15478.8     August  15536.1      NaN       NaN
## 392  2024-09-01       15410.9  September      NaN      NaN       NaN
## 393  2024-10-01       15534.6    October      NaN      NaN       NaN
## 394  2024-11-01       15812.0   November      NaN      NaN       NaN
## 395  2024-12-01       15954.0   December      NaN      NaN       NaN
## 396  2025-01-01       15517.7    January      NaN      NaN       NaN
```
]
]
]

---
class: inverse, center, middle

# Classical Decomposition

---

## Classical Decomposition

.content-box-grey[
**Additive decomposition:** `$y_t = T_t+S_t+R_t = \hat{T}_t + \hat{S}_t + \hat{R}_t$`

**Multiplicative decomposition:** `$y_t = T_t\times S_t\times R_t = \hat{T}_t\times \hat{S}_t\times\hat{R}_t$`
]

- **Estimate** `$\hat{T}$` using `$(2\times m)$`-MA if `$m$` is even. Otherwise, estimate `$\hat{T}$` using `$m$`-MA

- **Compute de-trended series** using: (a) `$y_t - \hat{T}_t$` for additive decomposition, and (b) `$y_t/\hat{T}_t$` for multiplicative decomposition.

- Once de-trended, **we are left with** the `$S_t$` and `$R_t$` components. Specifically:  
  * **Additive model:** `$y_t-\hat{T}_t  = (\hat{T}_t+\hat{S}_t+\hat{R}_t) - \hat{T}_t = \hat{S}_t+\hat{R}_t$`   
  * **Multiplicative model:** `$\frac{y_t}{\hat{T}_t} = \frac{\hat{T}_t\times \hat{S}_t\times \hat{R}_t}{\hat{T}_t} = \hat{S}_t\times \hat{R}_t$`
  
.footnote[
<html>
<hr>
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**Source:** Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. <https://otexts.com/fpp3/>. Accessed on February 12, 2025. [Chapter 3.4 Slides](https://github.com/robjhyndman/fpp3_slides/blob/main/3-4-classical-decomposition.Rmd) 
]

---

## Estimating Seasonal Component

- The **seasonal component** can be estimated by **averaging** the **detrended series** for that season across all years.

- For example, if we have monthly data, we can estimate the seasonal index for January by averaging all the January values in the detrended series.  Do this for each month to get the seasonal index.

- **Adjust** the seasonal indices to ensure that that:  
    * for additive: `$S^{(1)}+S^{(2)}+\ldots+S^{(12)}=0$` 
    * for multiplicative: `$S^{(1)}+S^{(2)}+\ldots+S^{(12)}=m$`  
    
- The **seasonal component** is then obtained by **repeating** the **seasonal indices** for each year in the data.

.footnote[
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**Source:** Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. <https://otexts.com/fpp3/>. Accessed on February 12, 2025. [Chapter 3.4 Slides](https://github.com/robjhyndman/fpp3_slides/blob/main/3-4-classical-decomposition.Rmd) 
]

---

## Estimating the Remainder Component

Additive decomposition: `$\hat{R}_t = y_t - \hat{T}_t - \hat{S}_t$`

Multiplicative decomposition: `$\hat{R}_t = y_t / (\hat{T}_t\hat{S}_t)$`

---

## Classical Decomposition in Python (Ref)

---

## Classical Decomposition in Python (Code-Add)

.font90[

``` python
from statsmodels.tsa.seasonal import seasonal_decompose

retail_empl = (
  pd.read_csv("https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23ebf3fb&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1320&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=CEU4200000001&scale=left&cosd=1992-01-01&coed=2025-01-01&line_color=%230073e6&link_values=false&line_style=solid&mark_type=none&mw=3&lw=3&ost=-99999&oet=99999&mma=0&fml=a&fq=Monthly&fam=avg&fgst=lin&fgsnd=2020-02-01&line_index=1&transformation=lin&vintage_date=2025-02-12&revision_date=2025-02-12&nd=1939-01-01")
  .rename(columns = {"observation_date": "date", "CEU4200000001": "CEU4200000001"})
  .assign(
    date = lambda x: pd.to_datetime(x["date"]),
      )
  )

# Set the date as index and set the frequency of the data
retail_empl = retail_empl.set_index("date")
retail_empl = retail_empl.asfreq("MS")

# Decompose the time series
res = seasonal_decompose( retail_empl["CEU4200000001"], model = "additive")

# Extract the trend, seasonal, and residual components
res_df = pd.DataFrame({
  "observed": res.observed, "trend": res.trend, "seasonal": res.seasonal, "residual": res.resid
  })

# Plot the decomposed time series
fig = res.plot()
fig.set_size_inches(10, 4.75)
fig.tight_layout()
```

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/classical_decomp_py_add-1.png" width="960" style="display: block; margin: auto;" />
]

---

## Classical Decomposition in Python (Output-Add)

.footnote[
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**Comment:** You can have more control over the plot by plotting the data frame `res_df` directly. See this example from [StackOverflow](https://stackoverflow.com/a/55350230/10156153), while noting that their `decomposition` is equivalent to our `res` object. You can also convert the data into a long format using `pd.melt`, then `.reset_index()` to convert date into a column, and plot it using `sns.relplot`. 
]

---

## Classical Decomposition in Python (Code-Mult)

.font90[

``` python
from statsmodels.tsa.seasonal import seasonal_decompose

# Set the date as index and set the frequency of the data
retail_empl = retail_empl.set_index("date")
retail_empl = retail_empl.asfreq("MS")

# Decompose the time series
res = seasonal_decompose( retail_empl["CEU4200000001"], model = "multiplicative")

# Extract the trend, seasonal, and residual components
res_df = pd.DataFrame({
  "observed": res.observed, "trend": res.trend, "seasonal": res.seasonal, "residual": res.resid
  })

# Plot the decomposed time series
fig = res.plot()
fig.set_size_inches(10, 4.75)
fig.tight_layout()
```

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/classical_decomp_py_mult-5.png" width="960" style="display: block; margin: auto;" />
]

---

## Classical Decomposition in Python (Output-Mult)

.footnote[
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---

## Comments on the Classical Decomposition

- Estimate of trend is **unavailable** for first few and last few observations.  
- The **seasonal component repeats** from year to year, which may not be realistic (especially for long time-series).

- Since the trend-cycle is estimated using an average-based method, it is **not robust to outliers**.

- **Newer methods** are designed to overcome these problems.

.footnote[
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<hr>
</html>

**Source:** Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. <https://otexts.com/fpp3/>. Accessed on February 12, 2025. [Chapter 3.4 Slides](https://github.com/robjhyndman/fpp3_slides/blob/main/3-4-classical-decomposition.Rmd) 
]

---
class: inverse, center, middle

# STL Decomposition

---

## Seasonal-Trend Decomposition using LOESS (STL)

- **STL** is a very flexible method for decomposing time series.

- It can handle **any type of seasonality**, and allows for **non-constant** seasonal patterns.

- It is robust to **outliers**.

- However, it:   
  - **cannot** handle trading day or calendar adjustments, and  
  - it is only **additive** (take logs to get multiplicative decomposition).

---

## STL Decomposition in Python (Ref)

---

## STL Decomposition in Python (Code)

.font90[

``` python
from statsmodels.tsa.seasonal import STL

retail_empl = retail_empl.set_index("date")
retail_empl = retail_empl.asfreq("MS")

# Decompose the time series
*stl = STL(retail_empl["CEU4200000001"], seasonal = 13, robust = True)
*res = stl.fit()

# Extract the trend, seasonal, and residual components
res_df = pd.DataFrame({
  "observed": res.observed, "trend": res.trend, "seasonal": res.seasonal, "residual": res.resid
  })

# Plot the decomposed time series
fig = res.plot()
fig.set_size_inches(10, 4.75)
fig.tight_layout()
```

<img src="data:image/png;base64,#06_cma_and_decomposition_files/figure-html/stl_decomp_py-1.png" width="960" style="display: block; margin: auto;" />
]

---

## STL Decomposition in Python (Out)

---

## STL Decomposition vs Classical Decomposition

.black[.bold[In two minutes, examine the differences between the output figures from the classical decomposition (additive) and the STL decomposition.]]

.can-edit.key-activity7[

- Edit me  
  
- ... 
  
- ...

- ...

]

---

class: inverse, center, middle

# Recap

---

## Summary of Main Points

By now, you should be able to do the following:

- Describe and compute centered moving averages

- Estimate trend-cycle via moving averages

- Perform classical decomposition (trend-cycle, seasonal, residual/remainder)

- Understand STL / MSTL as alternatives to classical decomposition

---

## 📝 Review and Clarification 📝

1. **Class Notes**: Take some time to revisit your class notes for key insights and concepts.
2. **Zoom Recording**: The recording of today's class will be made available on Canvas approximately 3-4 hours after the session ends.
3. **Questions**: Please don't hesitate to ask for clarification on any topics discussed in class. It's crucial not to let questions accumulate.

---

## 🎯 Assignment 🎯

- In preparation for the assignment, you are encouraged to thoroughly read the following:  
  - [MSTL API Reference](https://www.statsmodels.org/devel/generated/statsmodels.tsa.seasonal.MSTL.html) from the statsmodels package.  
  - [Multi-Seasonal Time Series Decomposition using MSTL in Python](https://towardsdatascience.com/multi-seasonal-time-series-decomposition-using-mstl-in-python-136630e67530/), a Towards Data Science Blog, which demonstrates how the MSTL method works and can be implemented on a similar dataset.

- Skim through the [MSTL Research Paper](https://arxiv.org/pdf/2107.13462).

- Once you read the references and went over our class notes, you are now ready to examine and complete [Assignment 04](https://miamioh.instructure.com/courses/240425/assignments/3256295).